CENTRE FOR APPLIED MACROECONOMIC ANALYSIS

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1 CENTRE FOR APPLIED MACROECONOMIC ANALYSIS The Australian National University CAMA Working Paper Series May, 2005 SINGLE SOURCE OF ERROR STATE SPACE APPROACH TO THE BEVERIDGE NELSON DECOMPOSITION Heather M. Anderson The Australian National University Chin Nam Low Monash University Ralph Snyder Monash University CAMA Working Paper 11/2005

2 Single Source of Error State Space Approach to the Beveridge Nelson Decomposition Heather M. Anderson Australian National University Canberra, ACT, 0200 AUSTRALIA Chin Nam Low Monash University Clayton, Victoria 3800 AUSTRALIA Ralph Snyder Monash University Clayton, Victoria 3800 AUSTRALIA Abstract A well known property of the Beveridge Nelson decomposition is that the innovations in the permanent and transitory components are perfectly correlated. We use a single source of error state space model to exploit this property and perform a Beveridge Nelson decomposition. The single source of error state space approach to the decomposition is computationally simple and it incorporates the direct estimation of the long-run multiplier. Keywords: Beveridge Nelson decomposition; Long-run multiplier; Single source of error; State-space models. JEL classification: C22, C51, E32 *Correspondence: Heather.Anderson@anu.edu.au

3 1. Introduction Two stylized facts associated with most macroeconomic time series are long run growth and recurrent fluctuations around the growth path. This has often led to exercises which decompose macroeconomic series into permanent and transitory components, where the permanent component represents long run growth or trend in the economy, and the transitory component represents the business cycle. Canova (1998) outlines several different methods for trend/cycle decomposition, focussing on differences between "economic" and "statistical" procedures. One decomposition that has attracted considerable attention in the applied macroeconomics literature is that first proposed by Beveridge and Nelson (BN) (1981). They defined the permanent component of an integrated time series as the level of the long run forecast (minus the deterministic trend, if any), and the transitory component as the difference between the present level and the permanent component. By-products of this definition are that the same innovation drives the BN permanent and transitory components, so that innovations in these components are perfectly correlated. Another well known decomposition is the unobserved components (UC) decomposition advocated by Harvey (1989), in which innovations in the trend and cyclical components have zero correlation by assumption. This decomposition is popular in forecasting circles, but it is also used in applied macroeconomic analysis. An alternative UC forecasting approach advocated by Ord, Koehler and Snyder (1997) is the class of state space models with a single source of disturbance. In these latter models, innovations in the unobserved state components are perfectly correlated because they are driven by the same disturbance. It is this similarity with the BN property that motivates our use of a single source of error (SSOE) state space forecasting approach to undertake a BN decomposition. Our suggested SSOE approach to the BN decomposition focusses on the well known fact that BN permanent and transitory components correspond to those from a UC trend and cycle decomposition with perfectly correlated disturbances. Previous approaches, including those by Miller (1988), Newbold (1990) and Morley (2002) have focussed on overcoming difficulties associated with truncating infinite sums in the BN permanent 2

4 component. We avoid truncation issues by working with an (equivalent) BN decomposition outlined in Stock and Watson (1988). Our SSOE approach also delivers a direct estimate of the long-run multiplier, which is often used as a measure of persistence in applied macroeconomic analysis. 2. Beveridge Nelson Decomposition Assume that y t is a I(1) variable with a Wold representation given by y t = µ + γ (L) ε t, (2.1) where µ is the drift, γ(l) is a polynomial in the lag operator L with γ(0) = 1 and Σ i=0 γ i <, andε t is an iid 0,σ 2 one-step-ahead forecast error of y t.usingthewell known identity that γ(l) =γ(1) + (1 L)γ (L), we can rewrite (2.1) as y t = µ + γ (1) ε t +(1 L)γ (L)ε t, (2.2) or equivalently as µ y t = (1 L) + γ (1) ε t (1 L) + γ (L)ε t = τ t + c t. (2.3) The BN permanent component is given by τ t = µ (1 L) + γ (1) ε t (1 L), while the transitory component is c t = γ (L)ε t, showing the perfect correlation between innovations to τ t and c t. The expression for τ t can be rewritten as τ t = µ + τ t 1 + γ (1) ε t, (2.4) and if γ(l) is an ARMA(p, q) process with γ(l) = θ(l) φ(l) = 1+θ 1L+...+θ ql q 1+φ 1 L+...φ p L p then c t = φ p(l)c t + ψ n(l)ε t +(1 γ(1))ε t, (2.5) where φ p(0) = ψ n(0) = 0 and the orders of φ p(l) and ψ n(l) are p and n with n max(p 1, q 1). Economists are often interested in the long-run multiplier γ (1), which measures the long-run effect of a shock ε t on y t. Here, we note from (2.4) and (2.5) that γ (1) determines the relative size of (contemporaneous) innovations to each component. 3

5 3. Single Source of Error State Space Models The single source of error state space model proposed by Snyder (1985) is y t = β 0 x t 1 + e t (3.1) with x t = Fx t 1 + αe t, (3.2) where (3.1) and (3.2) are measurement and system equations. The k-vector x t represents the unobserved state of the underlying process at time t, e t is an i.i.d. 0,σ 2 innovation, and α, β and F are respectively (k 1), (k 1) and (k k) matrices. The key feature of this specification is that all equations are driven by the same innovation. Details regarding the stability and estimation of this model can be found in Snyder (1985) and Ord, Koehler, and Snyder (1997). 4. Single Source of Error State Space Approach to BN Decomposition AcomparisonoftheBNdecompositioninSection2withtheframeworkinSection3 reveals that the former fits into the SSOE framework if one substitutes (2.4) and (2.5) into y t = τ t + c t to obtain a measurement equation given by y t = µ + τ t 1 + φ p(l)c t + ψ n(l)ε t + ε t, (4.1) and then uses (2.4) and (2.5) as system equations (so that α in (3.2) becomes α =(γ(1), (1 γ(1)))). These system equations are similar to those in standard UC decompositions, but they identify the unobserved components by assuming that the trend and cycle innovations have perfect correlation, rather than zero correlation. Our SSOE approach uses the perfect correlation between the contemporaneous innovations in (2.4) and (2.5) to parameterize the BN decomposition, and this allows direct estimation of γ(1). In contrast, Morley s (2002) state space approach is based on a parameterization of y t that facilitates the direct estimation of each ARMA coefficient. Following the convention of calling the two BN components the trend and the cycle, the parameter of interest in empirical studies of (the logarithms) of output 4

6 is typically γ (1), which measures the long run percentage increase in GDP resulting from a 1% shock in GDP in one quarter. If γ (1) < 1 then the trend and cycle will have perfect positive correlation and both components will share in the variation of the data. However, if γ (1) > 1, then innovations in the trend and cycle will have perfect negative correlation, and τ t will be more variable than y t. Proietti (2002) and others have questioned whether one should call τ t a trend when it is more volatile than output itself, but Morley et al (2003) (who found that γ (1) > 1 forrealusgdp) observed that a shock to output can shift the trend so that output is behind trend until it catches up. Thus it is quite reasonable for trend innovations to be negatively correlated with cycle innovations and for trend innovations to be more variable than output innovations. 5. Applications We illustrate the use of the SSOE approach to performing BN decompositions for ARIMA(2,1,2) models of the logarithms of real output for the United States, the United Kingdom and Australia. The US model coincides with one used by Stock and Watson (1988) when studying the contribution of the trend component to real US GNP, and we examine corresponding models for the UK and Australia to demonstrate the relative contribution of trends in other countries. The permanent and transitory components of the ARIMA(2,1,2) model are τ t = µ + τ t 1 + γ (1) ε t and c t = φ 1 c t 1 φ 2 c t 2 + θ 1 ε t 1 +(1 γ (1))ε t, whichleadtoassoestatespaceformgivenby h y t = µ + i 1 φ 1 φ 2 θ 1 τ t 1 c t 1 c t 2 ε t 1 + ε t and 5

7 τ t c t c t 1 ε t = µ φ 1 φ 2 θ τ t 1 c t 1 c t 2 ε t 1 + γ (1) 1 γ (1) 0 1 ε t. We use quarterly data for USA GNP (1947:1 to 2003:1), for UK GDP (1960:1 to 2003:1) and for Australian GDP (1979:3 to 2003:3). Our estimates based on the state space representations correspond with direct maximum likelihood estimation of the ARIMA model, and they satisfy the relevant stability conditions. We remove θ 1 in theukmodel,afterfinding that it is statistically insignificant. Measures of trend are reported in Table 1, while the implied transitory components are illustrated in Figure 1 together with reference recessions published by the NBER and the ECRI. While there are often pronounced declines in the transitory components around NBER/ECRI peak to trough episodes, there are also clear differences between BN output cycles and conventional business cycles. This is unsurprising, given that each type of cycle has been constructed to serve different purposes, and has been based on different information sets. Our primary measure of trend is γ (1) ( 1+θ 1+θ 2 1+φ 1 +φ in terms of the ARMA coefficients 2 for y t ), which is Campbell and Mankiw s (1987) persistence measure that predicts the long run increase in output resulting from a 1% shock in output in one quarter. We find that γ (1) > 1 for all countries, implying strong persistence. This result also implies negative correlation between innovations to trend and cycle. To provide an alternative perspective on trend, Table 1 also reports Stock and Watson s (1988) R 2 measure of the fraction of the variance in the quarterly change in real output that can be attributed to changes in trend. These R 2 measures indicate that trend makes relatively lower contributions in the USA and Australian cases than it does in the UK. 6. Conclusion This paper demonstrates the use of a single source of error state space approach to compute the permanent and transitory components of the BN decomposition. This 6

8 approach exploits the BN property that the two components are perfectly correlated, and it allows direct inference on the long-run multiplier γ (1) as opposed to indirect inference based on the ARIMA coefficients. 7. References Beveridge, S., and C. R. Nelson (1981). A New Approach to Decomposition of Economic Time Series into Permanent and Transitory Components with Particular Attention to Measurement of the Business Cycle. Journal of Monetary Economics, 7, Campbell, J. and N.G.Mankiw (1987). Permanent and Transitory Components in Macroeconomic Fluctuations, The American Economic Review, 77, Canova, F. (1998). Detrending and Business Cycle Facts, Journal of Monetary Economics, 41, Economic Cycle Research Institute, Harvey, A. C. (1989). Forecasting, Structural Time Series Models and the Kalman Filter, Cambridge University Press. Miller, M. (1988). The Beveridge-Nelson Decomposition of Economic Time Series - Another Economical Computational Method, Journal of Monetary Economics, 21, Morley, J. C. (2002). A State-Space Approach to Calculating the Beveridge-Nelson Decomposition, Economics Letters, 75, Morley, J. C., C. R. Nelson, and E. Zivot (2003). Why are the Beveridge-Nelson and Unobserved Components Decompositions of GDP so Different?, The Review of Economics and Statistics, 85(2), National Bureau of Economic Research, Newbold, P. (1990). Precise and Efficient Computation of Beveridge-Nelson Decomposition of Economic Time Series, Journal of Monetary Economics, 26, Ord, J.K., A. B. Koehler, and R. D. Snyder (1997). Estimation and Prediction for a Class of Dynamic Nonlinear Statistical Models, Journal of the American Statistical Association, 92, 440, Proietti, T. (2002), Forecasting with Structural Time Series Models, in Clements and Hendry (eds), A Companion to Economic Forecasting, (Prentice-Hall). 7

9 Snyder, R. D. (1985). Recursive Estimation of Dynamic Linear Models, Journal of the Royal Statistical Society, Series B, 47, Stock, J. H., and M. W. Watson (1988). Variable Trends in Economic Time Series, Journal of Economic Perspectives, 2(3), Table 1: Measures of trend in log real GNP/GDP Country Long-run multipliers (γ(1)) and standard errors USA (0.1459) UK (0.1587) Australia (0.0460) R R 2 statistics are obtained by regressing the quarterly change in GNP against the change in BN trend Figure 1: Transitory components for ARIMA(2,1,2) models of output for USA, UK and AUSTRALIA USA UK AUST RALIA Shaded areas on the graphs indicate peak to trough episodes (recessions) recorded by the NBER (USA) and the ECRI ( UK and Australia) 8